Theorem dmmpt

Description: The domain of the mapping operation in general. (Contributed by NM, 16-May-1995) (Revised by Mario Carneiro, 22-Mar-2015)

	Ref	Expression
Hypothesis	dmmpt.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
Assertion	dmmpt	⊢ dom 𝐹 = { 𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V }

Proof

Step	Hyp	Ref	Expression
1		dmmpt.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
2		dfdm4	⊢ dom 𝐹 = ran ◡ 𝐹
3		dfrn4	⊢ ran ◡ 𝐹 = ( ◡ 𝐹 “ V )
4	1	mptpreima	⊢ ( ◡ 𝐹 “ V ) = { 𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V }
5	2 3 4	3eqtri	⊢ dom 𝐹 = { 𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V }